In a large bowl, whisk together flour, ground ginger, baking soda, and salt; set aside.

Heat butter in a small pot over medium-low heat until just melted, about 2 minutes. Remove from heat and stir in granulated sugar, molasses, and fresh ginger. Set aside until mixture is just warm and then whisk in egg. Add butter-molasses mixture and apricots to reserved flour mixture, stir until just barely combined, and then fold in chocolate. Cover and chill until firm, about 1 hour.

Preheat oven to 350 degrees F. Line 2 large baking sheets with parchment paper; set aside. Place raw sugar in a small bowl. To form each cookie, shape 1/2 tablespoon dough into a ball, and then roll in raw sugar to coat. Transfer to prepared baking sheets, spacing cookies about 2 apart. Bake until just cracked on top, 8 to 10 minutes. (To make larger cookies, use 1 tablespoon dough for each ball and bake 10 to 12 minutes.) Set aside to cool, then dust with powdered sugar if desired. Makes 8 dozen small (or 4 dozen large) cookies.